Finite Element Analysis of Intermediate Crack Debonding in ... · Draft 1 1 Finite Element Analysis of Intermediate Crack Debonding in FRP 2 Strengthened RC Beams 3 Michael Cohen1,

Draft

Finite Element Analysis of Intermediate Crack Debonding in

FRP Strengthened RC Beams

Journal: Canadian Journal of Civil Engineering

Manuscript ID cjce-2017-0439.R2

Manuscript Type: Article

Date Submitted by the Author: 09-Apr-2018

Complete List of Authors: Cohen, Michael; University of Waterloo, Civil & Environmental Engineering Monteleone, Agostino; AMTEC Engineering Ltd Potapenko, Stanislav; University of Waterloo, Civil & Environmental Engineering

Keyword: Debonding, Fracture, Interface, Finite Element Analysis, ABAQUS

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Canadian Journal of Civil Engineering

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Finite Element Analysis of Intermediate Crack Debonding in FRP 1

Strengthened RC Beams 2

Michael Cohen1, Agostino Monteleone

2, Stanislav Potapenko

3 3

1 PhD Candidate, Dept. of Civil and Environmental Engineering, University Of Waterloo, Waterloo, ON, Canada 4

(corresponding author). E-mail: [email protected] 5

2 AMTEC Engineering Ltd., Etobicoke, Ontario, Canada. E-mail: [email protected] 6

3 Dept. of Civil and Environmental Engineering, University Of Waterloo, Waterloo, ON, Canada. E-mail: 7

[email protected] 8

Word Count: 7474 9

Abstract 10

The performance of the fibre reinforced polymer (FRP) to reinforced concrete (RC) interface is 11

vital to ensure desired design capacity. Without proper understanding of the interfacial behaviour 12

it is impossible to develop an effective, efficient and rational bonding technique. This paper 13

presents the results of a comprehensive numerical investigation aimed to assess and better 14

understand the debonding behaviour caused by different types of intermediate flexural crack 15

distributions in FRP-RC strengthened beams. The model is based on damage mechanics 16

modelling of concrete, a bilinear bond-slip relationship with softening to represent the interface, 17

and a discrete crack approach to simulate crack propagation. The model also highlights how 18

crack propagation and debonding is affected by the rate of change of moment. It is shown that 19

the variation of crack spacing and rate of change of moment can significantly affect debonding 20

crack propagation and strain development in the internal and external reinforcement, which 21

directly influences debonding load. 22

Key Words: Debonding, Fracture, Interface, Finite Element Analysis 23

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1 Introduction 24

In recent years, there has been an increased need for the strengthening or rehabilitation of 25

reinforced concrete (RC) structures due to the aging of infrastructure, demand for higher vehicle 26

loads, updates in design codes or inadequate original design. Fibre-reinforced polymers (FRP) 27

are now routinely considered as an effective method for such applications in the structural 28

engineering community due to their light weight and minimal labor and equipment requirements, 29

as compared to tradition methods. Furthermore, the initial high material costs of FRP can be 30

offset by the low installation and long-term maintenance costs. Numerous studies have been 31

conducted to prove the efficiency of utilizing FRP composites in structural elements (Nanni 32

1995; Arduini et al. 1997; Grace et al.1999; Ross et al. 1999; Brena et al. 2003). In spite of this, 33

industrial practitioners are still concerned about premature debonding of the plates before 34

reaching the desired strength or ductility. Unlike the failure modes of concrete crushing, shear 35

failure and FRP rupture, such debonding failure cannot be predicted by conventional RC theory. 36

Premature debonding initiates from the ends of the plate or from intermediate cracks (IC) in the 37

concrete. In practice, FRP debonding from the end of an IC sometimes is unavoidable and more 38

dominant despite careful surface preparation and good bond between FRP composites and 39

concrete. Currently our basic understanding of the mechanics of the bond and failure between the 40

FRP and concrete initiating from an IC is rather limited. 41

The presence of FRP bonded to the tension face of a RC beam will restrict but not prevent the 42

opening of an IC. Empirical studies have shown that FRP debonding may initiate from the 43

bottom of an IC near the middle of the span, where the bending moment and force in the FRP are 44

high. With an increase in load, the debonded zone grows and propagates towards the free-end of 45

the plate, leading to ultimate debonding failure of the strengthened structure. As of late, 46

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debonding failure and the presence of multiple cracks has started to attract attention in the 47

research community. In fact, FRP debonding failure under the presence of multiple cracks is 48

now being considered in design recommendations (fib 2001). However, the proposed model has 49

not yet been verified experimentally. 50

One of the advantages of finite-element models is their ability to capture quantities that could be 51

difficult to quantify experimentally, such as stress/slip concentrations and distributions along the 52

FRP-concrete interface. In addition, they provide insight into the effects of micro- and macro-53

cracking on the interfacial behaviour and they allow us to better obtain results which may vary 54

significantly from researcher to researcher, such as FRP strain (Yao et al. 2004). 55

A commonly accepted method of studying IC debonding by researchers and industrial 56

practitioners is to apply a direct shear test on the composite beam (Chen and Teng 2001). This 57

test involves pulling a FRP plate bonded to a concrete prism along the direction of its length to 58

determine the bond capacity of the strengthened specimen. Due to the shear lag phenomenon (i.e: 59

the uneven distribution of strains along the FRP-Concrete interface relative to the distance from 60

the loaded end), the bond capacity approaches a plateau value with increasing bond length. By 61

performing the direct shear test on members with different bond lengths, it is possible to 62

determine the effective bond length necessary to develop sufficient bond capacity between FRP 63

reinforcements and concrete members. While this test may be frequently employed, both an 64

empirical and analytical study performed by Teng et al. (2002) and Teng et al. (2005) prove 65

otherwise. The results from those studies indicate that debonding models with parameters 66

derived from the direct shear test significantly underestimate the maximum FRP strain in the 67

strengthened beam. These findings may be attributed to the presence of multiple secondary 68

cracks along the beam as the opening of cracks along the beam act to reduce relative sliding 69

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between concrete and FRP, and may lessen the rate of softening along the interface. Moreover, 70

the portion of FRP plate situated between two adjacent cracks is subjected to tension at both 71

cracks. In contrast to loading setting observed in conventional shear test where one end of the 72

FRP plate is loaded and the other end remains intact. The discrepancy between commonly 73

accepted practices and empirical evidence is just one example of why further investigations into 74

the mechanism that trigger FRP debonding from IC are needed in an attempt to improve the 75

efficiency of rehabilitation projects in civil engineering applications. 76

In this study, the results of a comprehensive numerical investigation are presented to illustrate the 77

complex interaction between the debonding strain and the flexural crack distribution. The model 78

is based on damage mechanics modelling of concrete and a bilinear bond-slip relationship with 79

softening behaviour to represent the FRP-concrete interfacial properties. A discrete crack 80

approach was adopted to simulate crack propagation through a nonlinear fracture mechanics 81

based finite element analysis. The model also highlights how crack propagation and debonding is 82

affected by moment gradient. The numerical simulations were validated against experimental 83

results and are capable of predicting behaviour observed during such tests. The results provide an 84

insight on the behaviour of a repair system that is gaining widespread use and will be of interest 85

to researchers and design engineers looking to successfully apply FRP products in civil 86

engineering applications. 87

2 Numerical Modelling 88

2.1 Geometric Discontinuities 89

Generally, there are two approaches to simulate fracture process in finite element modeling: a 90

continuum approach and a discrete approach. The continuum approach, commonly referred to as 91

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the “smeared crack approach”, treats fracture as the end process of localization and accumulation 92

of damage in continuum without creating a real discontinuity in the material. It is unable to trace 93

individual macro-cracks because it tends to spread crack motion over a region of the structure 94

rather than at localized points unless the characteristic dimension of the finite elements are 95

chosen small enough from the beginning of the analysis to accurately resolve the evolving 96

damage zone, as demonstrated by Lu et al. (2005). However, for real-life structures the 97

computation costs become excessive and impractical. Alternatively, the discrete crack approach 98

models a crack discretely as a geometric entity and was selected to simulate discontinuities due 99

to opening of dominant cracks, slipping of rebar, and debonding of the FRP sheet. This approach 100

has been modelled using zero-thickness interface elements for the aforementioned discontinuities 101

and their constitutive relationships are discussed in subsequent sections. 102

2.2 Concrete Modelling 103

The concrete damage plasticity model provided in ABAQUS (2007) is used to simulate the 104

nonlinear behaviour of concrete. To ensure that major cracking and crack growth do not occur in 105

locations other than where the predefined cracks are set, the compressive and tensile material 106

models were defined without limitation of strain capacity, as shown in Figure 1. The plasticity of 107

concrete material was modelled using the concrete damage plasticity model. In this plasticity 108

model, the damage of concrete was represented by a continuum (smeared) approach; meaning 109

that the model does not physically generate macro-cracks in concrete, instead the cracks are 110

indirectly accounted for by the way their presence affects the stress and material stiffness. The 111

crack process in concrete is not a sudden onset of new free surfaces but a continuous forming and 112

connecting of macro-cracks. Macro-cracking and crushing in concrete are represented by 113

increasing values of the hardening variables and these variables control the degradation of the 114

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C

elastic stiffness and the evolution of the yield surface. They are also closely related to the 115

dissipated fracture energy required to generate macro-cracks. Typically, the formation of macro-116

cracks is represented macroscopically as softening behavior of the material, which causes the 117

localization and redistribution of strain in a structure. One way to remove the evolution of 118

cracking is to modify the compressive hardening and tensile softening of the material model, 119

within the original smeared model. Cracking leads to softening behaviour of the material so if 120

any post-yield softening is removed from the model, the growth of cracking could be avoided in 121

the material thereby only permitting major cracking to occur within the predefined locations. 122

The discrete crack approach is adopted to account for “major” concrete cracking in the model by 123

predefining possible flexural cracks in the beam. The predefined cracks are divided into two 124

zones: traction-free and cohesive crack, as shown in Figure 2. A traction free crack physically 125

represents a “notch” that would be set in the concrete beam at the time of casting to ensure that 126

cracking initiates and evolves at a predefined location. No forces are transferred along this zone 127

and crack surfaces are completely separated. The force in the cohesive crack surface follows a 128

predefined response whereby a linear softening curve is employed to model tension softening of 129

concrete. The cohesive crack, preceding the formation of a real crack, is assumed to initiate if 130

tensile stresses reach the tensile strength of concrete, ��, whereas the real macro-crack is formed 131

when the energy required to create a unit area of crack is achieved. The area below the curve 132

represents the model I interfacial fracture energy, ��. Moreover, minor “hairline” cracks might 133

still occur in locations outside the predefined discrete cracks when internal stresses exceed the 134

tensile capacity of concrete, ��. However, the development of these minor cracks is restricted, as 135

opposed to the major discrete cracks defined earlier (Niu and Zhishen 2006). 136

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Unloading and reloading behaviours are modelled by a secant path, which means following a 137

straight line back to the origin upon unloading the stress. After cracking, no shear stress is 138

assumed to transfer along the crack surface. 139

2.3 Reinforcing Steel 140

The stress-stain relationship for reinforcing steel was modelled with the use of the classical metal 141

plasticity model, provided in ABAQUS (2007), using three linear segments. Three linear 142

segments are employed to account for yielding and strain hardening of the reinforcing steel. 143

Interface elements surround the reinforcing steel to account for slippage between rebar and 144

concrete. Interfacial slippage was defined according to the CEB-FIB model code (1993). 145

2.4 Fiber Reinforced Polymer Composites 146

The FRP composites are assumed to be linear-elastic. The rupture point in the stress-strain 147

relationship defines the ultimate stress and strain values for the FRP. These ultimate points were 148

originally adopted from the experimental program conducted by Brena et al. (2003). 149

2.5 FRP-Concrete Interface 150

The FRP-concrete interface refers to a thin layer of the adhesive and adjacent concrete within 151

which the relative deformation between FRP and concrete mainly happens, as revealed by 152

experimental studies (Yuan et al. 2004). This deformation occurs mainly due to shearing stresses 153

along the FRP-concrete interface (Mode II). In addition, interfacial normal (peel) stress also exits 154

at intermediate crack locations. However, these type of stresses were not considered in this 155

numerical study, because it is generally accepted that the debonding of the FRP resembles mode 156

II fracture behaviour as the adhesive layer transfers shear stresses from the concrete to the FRP. 157

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In strict sense (microscopic), any interface is naturally mixed-mode and the stress state within 158

the interface is very complicated (Hutchinson and Suo 1992). Only the debonding behaviour 159

within the adhesive layer may be like mode II fracture behaviour, whereas the general debonding 160

within the adjacent concrete layer may be associated with a concrete mode I fracture and mode II 161

shearing fracture behaviour. However, the latest research on this topic has shown that it is 162

acceptable to ignore the mode I fracture behaviour since it has a negligible effect on the overall 163

debonding process. The works that has led to this assumption are briefly outlined below: 164

• According to Rabinovitch and Frostig (2000), the concrete beam and FRP plate are in 165

contact at the vicinity of the flexural crack. This suggests that the normal interface stress 166

is compressive at this location, and therefore, doesn’t affect the debonding of the FRP- 167

concrete interface if friction is neglected. This is different from the normal stress at the 168

FRP plate end, which is tensile and plays a critical role in the plate-end debonding. 169

• Existing solutions proposed by Smith and Teng (2001) show that the normal stress has 170

little effect on the derivation of shear stress. 171

• By using a displacement discontinuity model, Wu et al. (2002) found that there exists a 172

linear correlation between mode I concrete and a mode II interfacial fracture energy 173

values for a given shearing fracture energy introduced on the crack surface. This means 174

that the overall debonding behaviour can be regarded as a mode II for intermediate 175

cracks. 176

Thus, it is considered to be an acceptable assumption that the IC debonding is treated as a mode 177

II fracture. 178

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On the other hand, the interface is modelled using the non-linear fracture mechanics approach 179

proposed by Lu et al. (2005) and shown in Figure 3. Its validity has been proven by numerous 180

researchers (Baky et al. 2007; Ebead et al. 2007). Under external load, interfacial shear stresses, 181

τ, develops along the interface due to the opening of the predefined cracks. The opening of the 182

predefined cracks is represented numerically as a finite slip, �, between the FRP plate and the 183

concrete beam. Initially, when the applied load is small and the interfacial stress, τ, is less than 184

the maximum interfacial stress, τmax, the interface is considered to be in its elastic stage. The 185

elastic stage ends when τ exceeds τmax, which signifies the onset of micro-debonding. As slip 186

continues to increase, the interfacial shear stresses reduce to zero, and full debonding initiates 187

and propagates along the interface, representing macro-debonding. Slip at the interface is given 188

by: 189

� = � − �� (1)

Where �is the horizontal displacement of the FRP element and �� is the horizontal 190

displacement of the concrete element. The interface is formulated as: 191

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� = �� if� ≤ �� −�(� �� − 1⁄ "if� > �� (2)

�� = 0.0195()�� ; � �� = 1.5()�� (3)

� = *+, -./0��12/4⁄ ; �� = 0.308()27�� (4)

() = 7(2.25 − 9� 9�⁄ )/(1.25 + 9� 9�⁄ ) (5)

where � �� is the maximum interfacial shear stress; �� is the slip at maximum interfacial shear 192

stress; � is a coefficient; �� is the interfacial fracture energy; and () is FRP width factor. 193

2.6 Structural Model 194

Due to symmetry, only half of beam was modeled in two-dimensions. The structural member is 195

broken down into finite elements to model the composite beam. Since more than one type of 196

materials and interfaces are considered in the analysis, different types of finite elements are 197

required to discretize the structure. Concrete is modelled using continuum elements; reinforcing 198

steel and FRP are modelled using one-dimensional beam elements; and predefined flexural 199

cracks, rebar-concrete interface and FRP-concrete interface are modelled using 200

cohesive/interface elements. Linear reduced-integration continuum elements are employed 201

throughout the analysis with a fine mesh for their ability to withstand severe distortion in 202

plasticity and crack propagation applications. Reinforcing steel is modeled as one-dimensional 203

beam elements discretely defined and superimposed onto the ‘host’ concrete element through the 204

use of beam (stringer) elements. The FRP composite is modelled as singly defined beam 205

elements attached to the concrete through interface elements. 206

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The interface elements are attached to the concrete substrate through the use of “tie constraints”. 207

At each end of the interface element, the interaction between the two nodes is represented by two 208

springs: shear spring with stiffness, <�=>�, and normal spring with stiffness, <>=>�. Load was 209

applied in the form of an imposed displacement at the top face of the specimen and a pin-support 210

was used to restrain the beam in the vertical direction. Plane strain condition was assumed 211

throughout the analysis, since during debonding failure the conditions at the crack tip are “neither 212

plane stress nor plane strain, but three-dimensional” (Anderson 1994; Coronado and Lopez 213

2007). IC debonding is a common failure mode in composites beams that exhibit flexural 214

behaviour when the shear span-to depth (a/d) ratio is approximately 2.5 or greater (Rosenboom 215

and Rizkalla 2008). Since the model has a shear span of 3.05, IC debonding behaviour is 216

expected. 217

2.7 Model Verification 218

The results of a comprehensive experimental study reported by Brena et al. (2003) have been 219

used to validate the original finite element model. To ensure the accuracy and effectiveness of the 220

numerical modelling approach followed in this study, the calibration of results was conducted on 221

a global (structural beam) level and a local (constitutive materials) level. Initially, the numerical 222

beam was calibrated against three different experimental specimens. At first, a control finite 223

element model (without CFRP) was validated against a corresponding experimental specimen to 224

the same author. Then, CFRP was added to the numerical model, but with the assumption of 225

perfect bond between the concrete substrate and the adjacent CFRP. Similarly, the simulation of 226

this model was calibrated with the tested beam, in order to emphasize the influence of accounting 227

for the appropriate interfacial mechanisms. Finally, the response of the main ABAQUS model in 228

this study (CFRP with cohesive zone approach) was compared to the real beam, and was found 229

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to replicate the original data very closely. Figure 4(a) through (c) illustrate the comparison 230

between each model and the experimental beam, in term of load-midspan deflection curve. 231

On the other hand, the validation of the FE model on local level was achieved by obtaining the 232

load-strain relationships of concrete, steel rebar, and CFRP laminate, and compared them with 233

experimental results (see Figure 5(a) through (c)). Good correlations between the numerical and 234

experimental results were found suggesting that the model has been adequately calibrated. The 235

numerical investigations have been designed to expand over previous experimental studies and 236

better understand the structural behaviour of crack induced debonding. The length of the FRP 237

plate was increased from 1322 to 2200 mm from the calibrated model to the model used for this 238

study, respectively. The modification was made to allow for more flexural cracks to span across 239

the entire length of the beam and study how the structural behaviour and debonding mechanisms 240

are influenced by multiple intermediate cracks, which develop under external loading, and are 241

virtually impossible to measure experimentally. The geometric properties of the model used in 242

this study is shown in Figure 6(a). 243

2.8 Mesh Convergence 244

A general check on the mesh density was investigated prior to the start of the analysis. Mesh 245

refinement was investigated for two models: a single discrete crack located under the load point 246

and multiple cracks distributed along the interface at 50 mm apart. The latter represents a case in 247

which convergence problems are expected to be most severe due to the complicated debonding 248

behaviour for such closely spaced cracks. Figure7 shows the results of the load-deflection 249

response using five levels of mesh refinement for the case of a single discrete crack located under 250

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the load point. It can be seen that the structural performance is overestimated with the use of the 251

coarse mesh, while refining the mesh leads to convergence of response. 252

While it appears that convergence is achieved by employing a fine structured mesh of element 253

size 5 mm by 5 mm, a closer inspection of the results in terms of interfacial shear stress versus 254

midspan deflection proves otherwise. Figure 7 demonstrates that the fine mesh employing 255

element sizes of 5 x 5 mm overestimates the interfacial shear stress response of the specimen (at 256

location immediately under the loading point), whereas mesh convergence is ultimately obtained 257

with the use of a finer mesh size of 3.5 x 3.5 mm. Overlooking the effect of interfacial shear 258

stress when selecting a mesh for the analysis may lead to inaccurate results along the interface as 259

the coarser mesh appears to be incapable of converging prior to macro-debonding. The finer 260

mesh seems to be able to accurately capture the complicated fracture behaviour involving 261

concrete cracking and interfacial debonding. This mesh was found to be suitable for the model 262

employing cracks spaced at 50 mm apart. 263

264

265

3 Numerical Analyses and Discussions 266

In this study, no attempt was made to consider the application of FRP composites to a pre-267

damaged RC structure. While in practice many cracks will have already formed with some 268

spacing before the application of FRP composites, a clear explanation on how these cracks may 269

affect the bond characteristics between concrete and FRP and ultimately influence the mode of 270

failure is currently unavailable. Without established empirical evidence on this topic, any finite 271

element model created to carry out such an analysis can potentially be contradicted. The cracking 272

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spacing, ��, is varied throughout the model to observe the how the spacing between cracks 273

affects the debonding mechanisms and efficiency of internal and external reinforcement. A 274

single localized crack model and five (5) crack spaced models (�� = 280, 125, 100, 75 and 50 275

mm) were considered as these values closely represent the real pattern in the tested specimens. 276

The following parameters are used in this study: b = 1000 mm, <�=>� = 160 MPa/mm (taken from 277

the experimental program), <>=>�= 0 MPa/mm, �?=4.5 MPa, ��=>�=0.5 N/mm, the width factor, 278

() was found to be 0.75, the �� value was 0.0516, and the value of the parameter � was 1.1015, 279

@�A= 230,000 MPa, ��= 3.5 MPa, @� = 26,500 MPa, ��B= 35 MPa, C�? = 200DD2 280

(∅16DD), C�� = 71DD2 (∅10DD), C��=AAH = 51DD2 (∅8DD), Reinforcement ratio, I, 281

is 0.0063. A schematic representation of the structural model is shown in Figure 6(b). It must be 282

noted that despite the presence of very few crack openings generated outside the constant 283

moment region of the beam, the cracks within the shear span were actually accounted for and 284

modelled as inclined cracks. The schematic representation of the FE model in Figure 6(b) is 285

intended to show the location of each crack, not the orientation of cracks. 286

287

3.1 Global Response 288

The effect of crack spacing is first analyzed based on the load-deflection behaviour of the FRP-289

strengthened beam. Figure 8 demonstrates that the stiffness and ultimate capacity of the model 290

decreases with the decrease in crack spacing. This can be attributed to the existence of more closely 291

spaced cracks reducing the rigidity of the structure as loading progresses. Single localized and 292

large crack space models produce similar stiffness and ultimate load, which are both greater than 293

the smaller crack space models. Table 1 lists key data throughout the parametric analysis and is 294

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referenced throughout Section 3. Once macro-debonding initiates, the debonding crack 295

propagates towards the end of the FRP sheet and the load would remain relatively constant until 296

final debonding failure (ultimate capacity). Subsequent to the initiation of macro-debonding, 297

debonding failure was reached earlier in models with more closely spaced cracks as opposed to 298

the larger crack spaced models by inspection of the deflection values for each model in Table 1. 299

The single localized crack model was able to deflect 18.9 mm after macro-debonding in 300

comparison to only 9.5 mm for the model with cracks spaced at 50 mm. This may be attributed 301

to the existence of more closely spaced cracks quickening the debonding propagation. 302

3.2 Interfacial Shear Stress Response 303

The above results can be explained by studying the interfacial behaviour of the analyses models. 304

For the case of the single localized crack predefined beneath the load, it was found that prior to 305

the initiation of flexural cracking there is no slip and therefore no shear stress at the FRP-306

concrete interface. With further loading, interfacial shear stresses develop along the interface 307

until micro-debonding initiates at a midspan deflection of 2.4 mm. At this point, micro-308

debonding occurs in the weaker concrete layer of the interface and high bond stresses develop 309

near the toe of the crack creating sliding between the concrete and FRP plate. The strain in the 310

plate is no longer equal to the strain in the adjacent concrete and the difference is defined as slip 311

strain. In order to accommodate the stress development, the FRP plate would require infinite 312

strains across the crack, which is not possible, and thus results in micro-cracking. This point is 313

illustrated in Figure 9 where the stress concentration at the toe of the crack reaches its maximum 314

value of 4.5 MPa. As loading progresses, the maximum interfacial shear stress shifts along the 315

beam in two directions: towards the support and midspan. This shift represents that the shear 316

capacity of the interface is reached and the development of the debonding crack, which is 317

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propagating along the soffit in two directions. To demonstrate the debonding propagation along 318

the interface Figure 9 captures the interfacial shear stress distribution at six (6) stages during the 319

analysis. It should be noted that the effective shear transfer length, Leff, required to attain the 320

ultimate load- carrying capacity may be regarded as 140 mm, which compares well with the 321

analytical solution proposed by Chen and Teng (2001). However, this parameter, Leff, is obtained 322

schematically from the strain distribution of FRP reinforcement, and is calculated as the distance 323

required to achieve full FRP bond capacity along the concrete interface. 324

The effects of crack spacing will be discussed by considering the 280 and 100 mm crack spacing 325

simulations. The case with cracks spaced at 280 mm represents a case in which flexural cracks 326

are well spaced along a beam (large crack spacing). It can be seen from Figure 10 that the 327

occurrence of a new crack adjacent to Crack 2 affects the interfacial shear stress distribution 328

since there is a change in slip direction. Now there exists a negative slip between neighbouring 329

cracks resulting in a change in direction of interfacial shear stress in order to maintain 330

equilibrium. This was not the case for the single localized crack. The development of negative 331

slip can be explained by examining the behaviour along the interface between Cracks 1 and 2. 332

When the slip at Cracks 1 and 2 are small, i.e. prior to the initiation of macro-debonding, shear 333

stresses along the interface develop at the toe of the flexural cracks, but in opposite directions. 334

The stresses at the right of Crack 1 develop towards the support and stresses to the left of Crack 2 335

develop towards the applied load. As load increases and macro-debonding initiates, the 336

debonding crack propagating towards the midpoint from Crack 2 restricts the debonding crack 337

that intends to propagate towards the support from Crack 1. The point at where these two 338

debonding cracks meet is referred to as the point of zero slip (Liu et al. 2007). From Figure 10, it 339

can be seen that the point of zero slip between Cracks 3 and 4 shifts towards the subsequent 340

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crack, Crack 4, as the slip at Crack 3 increases. This suggests that the uncracked concrete soffit is 341

gradually slipping towards Crack 4 and hence reducing the slip on the left of Crack 4. The 342

debonding crack that was developing from Crack 4 towards the applied load cannot propagate 343

any further and is restricted by the debonding crack propagating towards the support from Crack 344

3, indicating that the debonding crack propagating from the left of Crack 4 has closed up. Now 345

the debonding crack propagates in only one direction: towards the support. Since there are no 346

cracks predefined between Crack 4 and the plate-end, the debonding crack propagates rapidly 347

from Crack 4 to the plate-end once macro-debonding initiates causing debonding failure to occur. 348

Similar behaviour was observed between Cracks 2 and 3 but not between Cracks 1 and 2. This 349

may be attributed to the fact that the region between Cracks 1 and 2 lie in a constant moment 350

region, whereas the regions between Cracks 2 and 3 and Cracks 3 and 4 are located in a varying 351

moment region. As shown in Figure 10, to maintain equilibrium the point of zero slip in this 352

constant moment region was found to occur at the midpoint between Cracks 1 and 2, as opposed 353

to the varying moment region where the point of zero slip moves towards the next crack as slip 354

increases. It was found that the debonding crack in the constant moment regions will propagate in 355

both directions and will not close up like those in varying moment regions prior to debonding 356

failure. The effective shear transfer length was found to be 145 mm. 357

The case of cracks spaced at 100 mm along the interface is used to exemplify a case in which 358

cracks are closely situated along the soffit (i.e. small crack spacing). The debonding propagation 359

does not appear very smooth like that observed for the case of the single localized crack and large 360

crack spaced model (�� = 280 mm). This may be attributed to the existence of many cracks 361

along the interface spaced less than the effective shear transfer length. The effective shear 362

transfer length for the single localized crack was found to be 140 mm and predefining the cracks 363

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less than Leff appears to be complicating the debonding behaviour. The debonding crack 364

encounters resistance from the opposite direction near the adjacent cracks resulting in an 365

increased Leff of 215 mm as shown in Figure 11. Leff can be more easily determined by referring 366

to the FRP strain distribution along the interface, since the existence of very closely spaced 367

cracks complicates the debonding propagation. Similar trends were found for other small crack 368

spaced models of 125, 75, and 50 mm with Leff values of 190, 245, and 330 mm, respectively. It 369

appears as if more energy is required for the debonding crack to propagate through the flexural 370

cracks in small crack spaced models, which contribute to the increase in Leff. Inspecting the 371

deflection values in Table 1 suggests that the existence of multiple cracks spaced less than Leff 372

helps prolong the initiation of micro-debonding, rebar yielding, and macro-debonding. 373

The existence of a secondary crack and the spacing between adjacent cracks appears to have an 374

effect on the initial of debonding and rebar yielding, which occurred at earlier stages in models 375

with larger crack spacing (see Table 1). This may be attributed to the abrasion effect along the 376

interface, where additional work is required for debonding to propagate beyond secondary cracks 377

(Leung and Yang 2006). When a flexural crack opens under loading, longitudinal displacements 378

at the bottom of the beam increase. Due to the abrasion effect, the residual shear stress at any 379

point along the debonded zone decreases with interfacial relative sliding. As a result, the relative 380

displacement in the debonded zone is reduced and the interfacial shear stress will increase. In 381

another word, the presence of cracks was found to reduce the initiation of micro-debonding and 382

rate of interfacial softening. Consequently, the maximum force in the FRP was found to increase 383

with a decreased crack spacing, demonstrating the effectiveness of FRP rehabilitation in delaying 384

debonding and crack propagation. 385

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However, while some researchers have reported increases in ultimate load when crack spacing is 386

reduced in their finite element models (Niu and Wu 2005), such a phenomenon was not found to 387

occur in this study as the existence of more closely spaced cracks in the model greatly reduces 388

the rigidity of the structure. Subsequent to the initiation of macro-debonding, the rate of 389

debonding propagation was found to be increased in models with smaller crack spacing, 390

evidenced by the lower deflection values obtained before debonding failure in Table 1. 391

3.3 Internal Reinforcement Response 392

Figure 12 compares the rebar strain distribution along the beam for the single localized crack 393

model, and two (2) crack spacing of 125 and 50 mm at the initiation of macro-debonding and at 394

debonding failure. The results suggest that prior to macro-debonding the decrease in crack 395

spacing may be helpful to utilize the full strengthening effect of the internal reinforcing steel. 396

However, this may not be the case for prolonged loading as the results suggest that the internal 397

reinforcement becomes less effective in smaller crack spaced models subsequent to macro-398

debonding, as shown in Figure 12. This may be attributed to the fact that as the debonding crack 399

propagates along the interface, multiple flexural cracks open under loading creating longitudinal 400

displacements at the FRP-concrete interface. As the displacements increase, the flexural cracks 401

migrate upward and reduce the bond action between the concrete and rebar, creating slippage at 402

the concrete-rebar interface. In order to accommodate the load demand, the system now relies 403

more on the FRP reinforcement. Consequently, the maximum force in the rebar was found to 404

decrease with a decrease in crack spacing. This may also help to explain why lower ultimate 405

load was found in models with smaller crack spaced models than for the single localized crack or 406

large crack spaced models. 407

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3.4 External (FRP) Reinforcement Response 408

Once yielding occurs in the rebar, FRP strain increases at a much higher rate until interfacial 409

debonding occurs. For the case of the single localized crack and large crack space models, 410

debonding propagation along the interface occurs easily and remains constant once macro-411

debonding initiates as shown in Figure 13. This may be explained by considering cracks spaced 412

greater than Leff are less susceptible to the abrasion effect, as previously explained in Section 413

3.2, suggesting that the FRP sheet will produce similar strengthening results in cases where xc > 414

Leff. The single localized crack and large crack spaced model (�� = 280mm) produce similar 415

results following macro-debonding. However, with the case of small crack spacing, FRP strain 416

was found to continue to increase (at a comparatively lower rate) following macro-debonding, 417

suggesting that small crack spacing may be helpful to further utilize the strengthening effect of 418

the FRP sheet. This phenomenon is shown in Figure 13 for the xc = 100 mm model and may be 419

attributed to the abrasion effect, which is more prominent in the models with crack spacing less 420

than the Leff. In these models the FRP sheet continues to contribute to the load-carrying capacity 421

despite debonding at a particular location so long it is located within the effective shear transfer 422

length of a nearby flexural crack. Inspecting the FRP strain values listed in Table 1 supports this 423

observation as strain values are higher in smaller crack spaced models than larger spaced models 424

at debonding failure. It is interesting to note how the optimum crack spacing to achieve the 425

maximum plate strain is dependent on the crack spacing within the beam. As listed in Table 1 426

the FRP strain prior to failure is the greatest in the �� = 100 mm model, followed by the �� = 75, 427

125, and 50 mm models, respectively. Intuitively, one would expect the �� = 50 mm to have the 428

maximum FRP strain value at failure since it has the largest Leff of the group. However, the 429

existence of very closely spaced cracks in the model quickens the debonding propagation and 430

Page 20 of 42



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21

fails the structure earlier, thus not allowing the FRP sheet to be fully utilized for the �� = 50 mm 431

model. Moreover, Table 1 lists that the �� = 50 mm model was only able to deflect an additional 432

8.5 mm after macro-debonding initiates prior to debonding failure, in comparison to 13.7 mm and 433

18.9 mm for the xc = 100 mm and single crack spaced model, respectively. This suggests that 434

crack spacing and the amount of cracks along the beam influence the local stress and strain in a 435

beam. Furthermore, the �� = 100 mm model had the largest FRP strain increase after the 436

initiation of macro-debonding followed by the �� = 75, 50 and 125 mm models as listed in Table 437

1. 438

4 Conclusion 439

In this study, a detailed finite element model was developed to carry out a comprehensive finite 440

element investigation that provides useful information related to the mechanics of flexural IC 441

debonding in the FRP strengthened RC members. Numerical results presented in this paper 442

indicate that: 443

• Cracks spaced greater than the Leff produce similar ultimate load and FRP strain results 444

to the single localized crack model. 445

• Subsequent to the initiation of macro-debonding, the interface cracks spread towards the 446

end of the FRP more rapidly in models with smaller crack spacing. Ultimate capacity, 447

therefore, was reached earlier in models with more closely spaced cracks. 448

• The shear stress is both additive and subtractive on either side of each flexural crack due 449

to equilibrium of forces. The value will be a maximum in high moment regions where the 450

crack opening displacement has the greatest magnitude. 451

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• To maintain equilibrium in the constant moment region, a point of zero slip always 452

occurs at the mid-distance between two cracks and the debonding crack propagates in 453

both directions and will not close up easily like those in varying moment regions. 454

• The existence of the multiple cracks prolongs the initiation of micro-debonding, rebar 455

yielding, and macro-debonding. This may be attributed to the abrasion effect along the 456

interface. 457

• The internal reinforcement (rebar) becomes less effective in smaller crack spaced models 458

subsequent to the initiation of macro-debonding and the beam relies more on external 459

FRP reinforcement to contribute to its load carrying capacity460

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References 461

ABAQUS. 2007. User’s Manual version 6.7. Pawtucket (RI): Hibbit, Karlsson & Sorensen. 462

Anderson T. 1994. Fracture Mechanics: Fundamentals and Application. CRC Press, Boca Raton, 463

Fl. 464

Arduini M., Di Tommaso A., and Nanni A. 1997. Brittle Failure in FRP Plate and Sheet Bonded 465

Beams. ACI Structural Journal, 94(4): 363-70. 466

Baky H.A., Ebead U.A., and Neale K.W. 2007. Flexural and interfacial behaviour of FRP-467

strengthened reinforced concrete beams. Journal of Composites for Construction, 11(6): 629-468

638. 469

Brena S.F., Bramblett R.M., Wood S.L., and Kreger M.E. 2003. Increasing Flexural Capacity of 470

Reinforced Concrete Beams Using Carbon Fiber-Reinforced Polymer Composites. ACI 471

Structural Journal, 100(1): 36-46. 472

CEB Bulletin 213/214. 1993. CEB-FIB Model Code 90. International Federation for Concrete 473

Structures. Sprint-Digital-Druck Stuttgart, Lausanne, Switzerland. 474

Chen, J.F., and Teng, J.G. 2001. Anchorage Strength Models for FRP and Steel Plates Bonded to 475

Concrete. Journal of Structural Engineering, 127(7): 784-91. 476

Coronado C.A., and Lopez M.M. 2007. Damage Approach for the Prediction of Debonding 477

Failure on Concrete Elements Strengthened with FRP. Journal of Composites for 478

Construction, 11(4): 391-400. 479

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Ebead U.A., and Neale K.W. 2007. Mechanics of fibre-reinforced polymer-concrete interfaces. 480

Canadian Journal of Civil Engineers, 34: 367-377. 481

Fib Bulletin 14. 2001. Design and Use of Externally Bonded FRP Reinforcement for RC 482

Structures. International Federation for Concrete Structures. Sprint-Digital-Druck Stuttgart, 483

Lausanne, Switzerland. 484

Grace N.F., Sayed G.A., Soliman A.K., and Saleh K.R. 1999. Strengthening Reinforced 485

Concrete Beams Using Fiber Reinforced Polymer (FRP) Laminates. ACI Structural Journal, 486

96(5): 865-75. 487

Hutchinson, J.W., and Suo, Z. 1992. Mixed Mode Cracking in Layered Materials. Advances in 488

Applied Mechanics 29: 63-191. 489

Leung, C.K., and Yang, Y. 2006. Energy-Based Modeling Approach for Debonding of FRP 490

Plate from Concrete Substrate. Journal of Engineering Mechanics 132 (6), 583-593. 491

Liu I.S.T., Oehlers D.J., and Seracino R. 2007. Study of Intermediate Crack Debonding in 492

Adhesively Plated Beams. Journal of Composites for Construction, 11(2):175-183. 493

Lu X.Z., Ye L.P., Teng J.G., and Jiang J.J. 2005. Meso-Scale Finite Element Model for FRP 494

Sheets/Plates to Concrete. Engineering Structures, 27: 564-75. 495

Nanni A. 1995. Concrete Repair with Externally Bonded FRP Reinforcement. Concrete 496

International ACI, 17(6): 22-26. 497

Niu, H., Karbhari, V.M., and Zhishen, W. 2006. Diagonal Macro-Crack Induced Debonding 498

Mechanisms in FRP Rehabilitated Concrete. Composites Part B: Engineering, 37: 627-41. 499

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Niu H., and Wu Z. 2005. Numerical Analysis of Debonding Mechanisms in FRP-Strengthened 500

RC Beams. Computer-Aided Civil and Infrastructure Engineering, 20: 354-68. 501

Rabinovich O., and Frostig Y. 2000. Closed-Form High-Order Analysis of RC Beams 502

Strengthened with FRP Strips. Journal of Composites for Construction, 4(2): 65-74. 503

Rosenboom O., and Rizkalla S.H. 2008. Experimental Study of Intermediate Crack 504

Debonding in Fiber- Reinforced Polymer Strengthened Beams. ACI Structural 505

Journal, 105(1): 41-50. 506

Ross A., Jerome D.M., Tedesco J.W., and Hughes M.L. 1999. Strengthening of Reinforced 507

Concrete Beams with Externally Bonded Composite Laminates. ACI Structural Journal, 508

96(2): 212-21. 509

Smith S.T., and Teng J.G. 2001. Interfacial Stresses in Plated Beams. Engineering Structures, 23: 510

857-71. Taljsten B. 1997. Strengthening of Beams by Plate Bonding. Journal of Materials in 511

Civil Engineering, 9(4):206-12. 512

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Wu Z.S., Yin J., Ishikawa T., and Iizuka M. 2002. Interfacial fracturing and debonding failure 519

modes in FRP- strengthened concrete structures. In Proceedings of the Fourth Joint Canada-520

Japan Workshop on Composites, Vancouver, Canada, pp. 403-10. 521

Yao J., Teng J.G., and Lam L. 2004. Experimental Study on Intermediate Crack Debonding in 522

FRP- Strengthened RC Flexural Members. Advances in Structural Engineering, 11(4): 365-523

395. 524

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bonded joints. Engineering Structures, 36: 99-113. 526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

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List of Figure Captions 545

546

Fig 1. Strain hardening behaviour of concrete: (a) compression; (b) tension. 547

Fig. 2. Stress-relative displacement relationship of discrete cracking. 548

Fig. 3. Bond-behaviour of FRP-concrete interface. 549

Fig. 4: Global calibration of numerical model: (a) control beam; (b) CFRP Beam (Perfct Bond); 550

(c) CFRP Beam (CZM). 551

Fig. 5: Local calibration of numerical model: (a) concrete compressive response; (b) axial strain 552

in steel rebar; (c) axial strain in CFRP laminate. 553

Fig. 6: Schematic representation: (a) analysis model; (b) finite element model. 554

Fig. 7: Effect of mesh refinement for a single localized crack in terms of: (a) load versus 555

deflection; (b) interfacial shear stress under the loading point. 556

Fig. 8. Load versus deflection for single and large crack spacing. 557

Fig. 9. Interfacial stress distribution for the case of a single localized crack. 558

Fig. 10. Interfacial stress distribution for the crack spacing �� = 280mm. 559

Fig. 11. Interfacial stress distribution for the crack spacing �� = 100mm. 560

Fig. 12. Comparison of rebar strain distribution: (a) initiation of macro-debonding; (b) 561

debonding failure. 562

Fig. 13. Comparison of FRP reinforcement strain distribution: (a) single localized crack; (b) �� 563

= 100 mm. 564

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List of Table Captions 565

566

Table 1: Summary of effect of crack spacing analysis. 567

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29

568

Fig. 1: Strain hardening behaviour of concrete: (a) compression; (b) tension 569

570

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30

571

572

Fig. 2: Stress-relative displacement relationship of discrete cracking 573

574

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31

575

Fig. 3: Bond-behaviour of FRP-concrete interface 576

577

578

579

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(a) Conrol Beam

(b) CFRP Beam (Perfect Bond) (c) CFRP Beam (CZM)

Fig. 4: Global calibration of numerical model: (a) control beam; (b) CFRP Beam (Perfect Bond); 580

(c) CFRP Beam (CZM) 581

582

0

20

40

60

80

100

120

140

160

180

0 20 40 60

Load (kN)

Midspan Deflection (mm)

Experimental

Control Beam

0

20

40

60

80

100

120

140

160

180

0 20 40 60

Load (kN)


Experimental

CFRP Beam

(Perfect Bond)

0

20

40

60

80

100

120

140

160

180

0 20 40 60

Load (kN)


Experimental

CFRP Beam

(CZM)

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(a) Concrete response under compression (cylinder test)

(b) Axial strain in steel rebar (c) Axial strain in CFRP laminate

Fig. 5: Local calibration of numerical model: (a) concrete compressive response; (b) axial strain 583

in steel rebar; (c) axial strain in CFRP laminate 584

585

586

0

5

10

15

20

25

30

35

40

0.000 0.001 0.002 0.003 0.004

Stress (M

Pa)

Strain

Experimental

Numerical

0

20

40

60

80

100

120

140

160

180

0 0.005 0.01 0.015 0.02 0.025

Load (kN)

Rebar Axial Strain (at loading point)

Experimental

Numerical

0

20

40

60

80

100

120

140

160

180

0 0.005 0.01 0.015 0.02 0.025

Load (kN)

Rebar Axial Strain (at loading point)

Experimental

Numerical

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587

Fig. 6: Schematic representation: (a) analysis model; (b) finite element model 588

589

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35

590

Fig. 7: Effect of mesh refinement for a single localized crack in terms of: (a) load versus 591

deflection; (b) interfacial shear stress under the loading point. 592

593

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36

594

Fig. 8: Load versus deflection for single and large crack spacing. 595

596

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597

Fig. 9: Interfacial stress distribution for the case of a single localized crack. 598

599

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38

600

601

Fig. 10: Interfacial stress distribution for the crack spacing �� =280 mm. 602

603

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39

604

Fig. 11: Interfacial stress distribution for the crack spacing �� =100 mm. 605

606

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40

607

608

Fig. 12: Comparison of rebar strain distribution: (a) initiation of macro-debonding; (b) 609

debonding failure. 610

611

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612

613

Fig. 13: Comparison of FRP reinforcement strain distribution: (a) single localized crack; (b) �� = 614

100 mm. 615

616

617

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Table 1: Summary of effect of crack spacing analysis. 618

JK Init. of

Micro-

debonding

Yield load

Init. of

Macro -

debonding

Ultimate

load

FRP Strain

(Micro strain)

mm P kN ∆

mm

P

kN ∆

mm

P

kN ∆

mm

P

kN ∆

mm

∆HM�= ��N− ∆ ��A� O =�A� OP O ��A� OHM� %

Diff* SC 90.4 2.4 176.7 6.9 184.9 8.5 193.9 27.4 18.9 603.9 4500 5950 5950 0

280 90.8 2.7 165.7 7.1 178.9 9.0 194.3 25.6 16.6 664.4 4950 6188 6188 0

125 94.0 3.4 163.8 7.8 178 10.1 190.6 24.9 14.8 897.7 5135 6742 6924 2.7

100 110.7 4.5 168.5 8.1 179.8 10.3 193.5 24 13.7 971.7 5360 6372 7153 12.3

75 126.8 5.7 167 8.5 178.2 10.8 188.6 22.6 11.8 1286 5399 6453 7104 10.1

50 121.7 6.1 164.2 8.9 178.2 12 183.8 21.5 9.5 1368 4580 6354 6826 7.4

*Percent difference between macro and ultimate strain. 619

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